The stereotypical fangirl is a girl who loves a story/series for one or more of the male characters. They have existed for longer than you may think: the David DuchovnyEstrogen Brigade was a terrifying force on The X-Files' group on Usenet. The term has since widened beyond that stereotype to include girls who are merely fans of something or other.
It is not always a pejorative term but it can be among people who don't "get" a particular fandom. Most folks today use the term to mean nothing more than a female nerd. Someone describing herself as a fangirl is usually expressing an enthusiasm for the subject.
Also see: Estrogen Brigade, Squee
The lead character, Ichigo, begins the series as a fringe member of her eventual love interest's Instant Fanclub (he's the Kendo Team Captain). Both she and Minto grow to see their respective objects of affection as people rather than ideals.
Motomiya Jun of Digimon Adventure 02 is a living embodiment of this trope, going so far as to chase down the cool, aloof, and completely uninterested Ishida Yamato whenever within sight. Once it becomes clear that he's interested in Sora, she moves onto Jou's older brother. An audio drama seems to imply that they eventually have some kind of relationship: she visits him on Valentine's Day and gets introduced to his parents.
Naruto has Sakura and Ino who spent the first half of the series fangirling Sasuke Uchiha. Thankfully they got better and stopped. But in the second half of the series, which takes place after a Time Skip, we are introduced to Karin and Shiho, who both (especially the latter) alternate between calm and level-headed scientist-types to squealing fangirls for Sasuke and Shikamaru.
Also happens in the anime with the Starlights; particularly Ami, which shocks everyone.
Also in the anime, Minako turns out to be a fangirl of an idol singer - an important plot point in the end.
Ohno, Ogiue, and Ohno's friends Angela and Sue from Genshiken, though they're more realistic than most portrayals of fangirls (aside from Ohno and Angela's exaggerated proportions).
In her earliest appearances, Misato-chan seems to exist solely to fangirl Nana and the Black Stones, to the point of ignoring school, moving in with them to act as a fetch-and-carry girl, and even using a fake name to stump Yasu's attempted background check.
Konata Izumi from Lucky Star is a rare example of a different type of fangirl. She is not interested in shipping or attractive male fictional characters. She has some of the characteristics of a typical male Otaku, watching tons of anime, addicted to video games, and likes to cosplay. Some of this may be influenced by her father, who is also an otaku.
Hiyori Tamura, with her Yuri Fan and doujinshi drawing tendencies, is a more traditional example of this trope.
Emily Sevensheep of Mahou Sensei Negima! is a massive Nagi fangirl. In fact, many females in the Magic World are Nagi fangirls. There's a reason why being amongst the first 100 to join the official Nagi fanclub is considered a special thing. Hell, being in the first five digits is considered a big deal.
Her classmate Beatrix is a Rakan fangirl.
Ayaka runs the line between being a Negi fangirl and being a Shota Con. Most of the class think it's the second.
Kallen is Zero's biggest fangirl throughout most of Code Geass, complete with a Bodyguard Crush when Lelouch sees the true value of her devotion and puts her in charge of his personal defense.
Kaguya's an even bigger example.
Yumi from Maria-sama Ga Miteru gathers quite an extensive group of fan girls in the course of the series. She is the undisputable center of attention during the tea party she and Yoshino organize to find a suitable soer—much to Yoshino's chagrin.
Kanade from Candy Boy has her own personal fan girl in Sakuya. She doesn't seem too happy about most of the time, but she also doesn't really try to get rid of her, turning poor Sakuya into a Hopeless Suitor.
In Gundam SEED Destiny, Meer Campbell was so much of a fangirl for Lacus Clyne, she willingly became her Body Double and adopted her personality and, to a degree, her role.
Tomoyo is Sakura's fangirl in Cardcaptor Sakura, in addition to lovingher in other ways. Her obsession with filming all of Sakura's exploits even disturbs Sakura on occasion, and in an early episode, she gives an impromptu manifesto on Sakura's varied awesomeness to a boy who had just complimented Sakura on her athletic skill.
Mei Chan fangirls after Ed briefly in Fullmetal Alchemist, until she finally meets him and sees the shortness of him. She then BSODs briefly before bawling him out for 'toying with her heart', and then moves her affections to Al, who is suitably tall. No pun intended.
"Pipi" (the unnamed female character) from Saiyuki is a hard-core Kougaiji fan.
Nozomi of Chance Pop Session is a major Reika fangirl. The reason she joins the Akiba music school in the first place is for the chance to meet Reika. Even after having success as part of R-3, she still keeps her fangirl status.
In Project A-ko, C-ko has been A-ko's fangirl ever since A-ko rescued her from the wolf that was about to attack her.
Surprisingly, Yuki Nagato in the spin-off Melancholy of Haruhi-chan comes the closest ever to showing any emotion (thus far) by becoming the fangirl of an unnamed anime/game series.
In Yu-Gi-Oh!, Yugi has Rebecca Hawkins. Notable, she's only a Yugi fangirl, not to Yami Yugi.
Vivian Wong is a Yugi and Kaiba fangirl. She even kidnaps Yugi's grandpa, just because to get a date with our hero.
In Yu-Gi-Oh! GX, Rei Saotome (Blair Flannigan) was first introduced as a crazy Ryo Marufuji (Zane Truesdale) fangirl. Then, she turns into a Judai Yuki (Jaden Yuki) fangirl. She was also a Jun Manjoume (Chazz Princeton) fangirl for a minute, but then she remembers Judai. She's still hitting on Judai.
In Yu-Gi-Oh! 5Ds, Jack has Carly, Mikage/Mina and Stephanie. The three of them don't like each other.
This happens from time to time with the Bat-Family. Robin moved Spoiler to costumed adventuring (she later became an Ascended Fangirl), Oracle has a new one in Birds of Prey named Misfit, and back when she started being Batgirl could have been considered one herself.
Molly Hayes of Runaways used to be a major X-Men fangirl particularly when it came to Wolverine. Although once he showed up with Iron Man and Captain America (who had previously sent her to a foster home she ran away from) to arrest her friend Cloak, she decided he was a lot less cool in person.
A minor villainess named Virtuoso that appeared in the Secret Six series is a huge Fangirl of the deceased Fiddler and is the current holder of his violin. When Fiddler appeared in front of her as a Black Lantern, he could see that she was brimming with love. She claimed to be his biggest fan, gushed about his past work, and even offered to return his original violin.
Cars had Mia and Tia, the two little convertibles. After the race in the film's beginning, they "flash" their "headlights" at him. They later appear repainted light blue when their adoration has shifted to Chick Hicks.
Supernatural added an in-show fangirl Becky Rosen, who meets Sam and Dean in the first episode of season 5 and in the 9th episode of season 5, is a crazed fangirl who epitomizes the fangirl role as she can't seem to contain her excitement or stop touching Sam. And in one episode she even gave him a love potion, married him while he had it in him, took his pants, tied him to a bed, and almost went as far to sell her soul for more of the drug.
In Castle, Detective Kate Beckett is a devoted fan of the novels of Richard Castle, and has been since before she met him. Since after meeting him he has exasperated and infuriated her in equal measure by tagging along on her investigations, she would rather die than have him find out. As such, she's something of an aversion in that she manages to restrain any squeeing urges she might have remarkably well; this is primarily to avoid giving Castle the satisfaction, however, and they have been in danger of slipping out on occasion. It's all for naught, however, since Castle knows full well that she's a fangirl anyway and eagerly teases / flirts with her based on it regardless (not that he needs the excuse).
Some years ago, Kate was also a serious fan of a sci-fi show that was canceled after one season. She even used to go to conventions and do cos-play. The whole episode was also a hilarious meta-nod to Firefly.
The webshow within a show of iCarly has an obsessed and annoying fangirl named Mandy.
In Leverage, after Eliot plays the guitar once in public he gains a shocking amount of fangirls - in one scene they recognize him on the street and chase after him en masse. Later in the episode, Hardison spreads a rumor online, causing the fangirls to attack a car that they believe Eliot is in.
In the show Power Rangers Ninja Storm, specifically the episode I Love Lothor, Lothor stars in a tv show ala I Love Lucy, which sways the minds of anyone who watches the show to fall in fangirl love with him through subliminal signals, instantly creating an army of fangirls who swarm the studio with homemade signs and attack anyone who dares say anything against Lothor.
Penelope of The Amanda Show, who's obsessed with Amanda (and is played by Amanda).
Goes back a long way further than the David DuchovnyEstrogen Brigade . Seventies cop show Starsky & Hutch saw its own Estrogen Brigade volunteering in the devoted service of the two titular actors, most noticeably David "Hutch" Soul. Soul was propelled into a lucrative second career as a crooning pop star on the back of the series, with a long string of chart hits, lapped up by his adoring and swooning fan base. Who even now, in their fifties, still go to his gigs.
Consummate geek Jason Fox loved the first two Lord of the Rings movies. Return of the King was less enjoyable; the difference being that his teen sister Paige was now an Orlando Bloom fangirl.
Paige: I hope Orlando is in every shot of this movie! Please, please, please, let Orlando be in every shot! You read Return of the King, Jason — does Tolkien mention Orlando Bloom a lot? Wait, don't tell me! I want to be surprised!
Jason:(turns to someone off-panel) You, with the orc arrows — please shoot me.
A blog post eventually reveals that this is based on Bill Amend's wife's real-life fangirlism.
Betrayal at Krondor has the resident stoic badass Gorath run into a "an admirer" who's immediately all over him, causing him to do what anyone would do - flee.
It's hard to tell if the fangirls in the Final Fantasy VII prequel Crisis Core are a show of affection or not. They obsess over the four First Class SOLDIERs in the game - making fanclubs, sending e-mails about insipid details, and even arguing amongst themselves - just as their real life counter-parts do. There is even a side-quest that involves helping them which directly influences the final number of members in Zack's own fanclub.
Morgan LeFlay, Mighty Bounty Hunter(TM), from Tales of Monkey Island, is a rabid (and the only) Guybrush Threepwood, Mighty Pirate(TM), fangirl.
In The Last Story after a sidequest you'll get a few of them, with a fanclub...
In Solatorobo, if Red becomes a prize fighter, he'll get a fangirl named Toffee who almost confesses love to him before finding out that he's not the independent badass she thought he was. She later gets over it and starts crushing on him again - but by this point, she's made a name for herself and the move upsets her fanboys.
A truly terrifying example exists in And Shine Heaven Now. Around once a year the Hellsing characters get glomped by numerous chibi-fangirls who squee as they fly though the air and rant about how cool a character is after latching on. Even worse than this though is that the fangirls can also force people to act out their ships. In a recent storyline, the Major (a fanboy of War himself) turns this to his advantage. He gives a version of his 'I Love War' speech chock full of fandom terms to subdue the fangirls.
There was also the Something Positive storyline in which most of the attendees of a comic book convention were mauled to death (in variously creative ways) by a small army of angry fangirls.
Kanaya is this towards Rose in Homestuck, having read her walkthrough. Suffered a mild case of Warts and All after actually chatting, but is apparently getting over it very quickly now.
Also WV, after leading the Prospitians and Dersites to battle, he has many admirers, most of them female pawns from both sides.
On Scratched Earth, Jane is Betty Crocker's No.1 Fangirl and heiress to the company. Oddly enough, her Beta counterpart, Nanna, hated and feared the baroness, while her adoptive brother inherited the company.
uranianUmbra is a fangirl of the kids and the trolls. She even draws fanart of them (Which is drawn by an actual fanartist on Tumblr.) (Warning: link contains spoilers.)
Despite the fact that Fangirls make up a major part of the site, Gaia Online felt the need to mock fangirls in their lost Rejected Olympics Manga. The main event? Escape The Fan, in which a rabid fangirl is released from a box to chase down a Friendly Neighborhood Vampire and an attractive (and female) Dark Elf. Gratuitous Japanese and hilarity ensues, culminating with the fangirl getting a nosebleed during an attempted (same sex) Kiss of the Vampire, and crying about her OTP when said kiss is interrupted.
In the Whateley Universe, high school girl Fey has her own fan club chock full of fangirls who are also superpowered mutants. Not only does this mortify her, but her friends tease her about it.
Official Fanfiction Universities deal with... curing ...fangirls. Sample students include:
Nara from the Myst OFU, who calls for "Eedrah my love" in his wife's presence, and later writes a Fan Fic that brings a Character Replacement into being.
The Nostalgia Critic fits the stereotype more than he does the male version. When he likes something, he'll glee over it and go into squee levels of excitement. More often than not he'll get crushed by the film making a bad move but it's still cute to see.
We Are Our Avatars: Though normally stoic, Caroline Parton/Athena gets a fangasm whenever a Bishounen from a video game she likes comes into the picture. She immediately asks for their autographs and/or gives them a hug.
In Ultra Fast Pony, Pinkie Pie is a fangirl for The Lord of the Rings films. She frequently works LOTR references into her conversations, she wrote a massive Self-Insert Fic (where her author avatar ends up in a love triangle with Bill the pony and Shadowfax), and her main reason for going to the Grand Galloping Gala was to share this fic with as many people as possible.
Sierra from season three of Total Drama is, as the series puts it, an "Uber-fan", she mentions that she's created fansites for each and every character and knows their eatings habits, hopes and fears and even their dental records. Needless to say, she creeps everyone out.